Linear Algebra, Theory And Applications, 2012a

200 VECTOR SPACES AND FIELDS
Example 8.1.2 Let Ω be a nonempty set and let V consist of all functions defined on Ω
which have values in some field F. The vector operations are defined as follows.
(f + g)(x) = f (x)+g (x)
(αf)(x) = αf (x)
Then it is routine to verify that V with these operations is a vector space.
Note that F n actually fits in to this framework. You consider the set Ω to be {1, 2, ··· ,n}
and then the mappings from Ω to F give the elements of F n . Thus a typical vector can be
considered as a function.
Example 8.1.3 Generalize the above example by letting V denote all functions defined on
Ω which have values in a vector space W which has field of scalars F. The definitions of
scalar multiplication and vector addition are identical to those of the above example.
8.2 Subspaces And Bases
8.2.1 Basic Definitions
Definition 8.2.1 If {v 1 , ··· , v n }⊆V, avectorspace,then
{ n
}
∑
span (v 1 , ··· , v n ) ≡ α i v i : α i ∈ F .
Asubset,W ⊆ V is said to be a subspace if it is also a vector space with the same field of
scalars. Thus W ⊆ V is a subspace if ax + by ∈ W whenever a, b ∈ F and x, y ∈ W. The
span of a set of vectors as just described is an example of a subspace.
Example 8.2.2 Consider the real valued functions defined on an interval [a, b]. Asubspace
is the set of continuous real valued functions defined on the interval. Another subspace is
the set of polynomials of degree no more than 4.
Definition 8.2.3 If {v 1 , ··· , v n }⊆V, the set of vectors is linearly independent if
i=1
n∑
α i v i = 0
i=1
implies
α 1 = ···= α n =0
and {v 1 , ··· , v n } is called a basis for V if
span (v 1 , ··· , v n )=V
and {v 1 , ··· , v n } is linearly independent. The set of vectors is linearly dependent if it is not
linearly independent.